Banks, S.P. and Moser, A. (1994) Nonlinear Control Design in the Frequency Domain. UNSPECIFIED. ACSE Research Report 528 . Department of Control and Systems Engineering
Abstract
A first step is made towards a complete generalization of the classical linear frequency domain theory of feedback control. First, the theory of partial fraction expansions is extended to multi-dimensional complex rational functions. As expected, the theory is now much more complicated and requires the use of ideal theory and notions from algebraic geometry. It turns out that, as in the linear case, the coefficients of the expansion (which are now polynomials) are obtained by "removing the given singularity" and evaluating on the single variety, i.e. evaluating the rational function modulo the singularity in the coordinate ring of the singularity. Multi-dimensional residue theory is based on the use of homology groups of the space ( in fact, the compactified version S2n) minus the singularity version T. We shall show that the inversion of the n-dimensional Laplace transform can be performed by finding a homology basis pf Hq (S2n\T), and a dual basis of Hr-1 (TU {oo}), where r+q=n. This will reduce the computation in many cases to a simple application of the n-dimensional version of Cauchy's theorem. The use of the theory in feedback control design is given with a particular study of a simple second-order bilinear system. We shall define an imlicit closed-loop transfer function (which is nonseparable) and then apply norm inequalities in the time domain to complete the stability analysis.
Metadata
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Copyright, Publisher and Additional Information: | The Department of Automatic Control and Systems Engineering research reports offer a forum for the research output of the academic staff and research students of the Department at the University of Sheffield. Papers are reviewed for quality and presentation by a departmental editor. However, the contents and opinions expressed remain the responsibility of the authors. Some papers in the series may have been subsequently published elsewhere and you are advised to cite the later published version in these instances. |
Keywords: | Nonlinear Control Design; Frequency Domain; Residues, Multi-Dimensional Laplace Transform. |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Automatic Control and Systems Engineering (Sheffield) > ACSE Research Reports |
Depositing User: | MRS ALISON THERESA BARNETT |
Date Deposited: | 16 Jul 2014 11:56 |
Last Modified: | 28 Oct 2016 11:14 |
Status: | Published |
Publisher: | Department of Control and Systems Engineering |
Series Name: | ACSE Research Report 528 |